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SAFETY RELIEF VALVE SIZING: COMPARISON OF
TWO-PHASE FLOW MODELS TO EMPIRICAL DATA
A Senior Honors Thesis
By
PAUL ROBERT MEILLER
Submitted to the Office of Honors Programs & Academic Scholarships
Texas A&M University In partial fulfillment of the requirements
For the Designation of
UNIVERSITY UNDERGRADUATE RESEARCH FELLOWS
April 2000
Group: Engineering
SAFETY RELIEF VALVE SIZING: COMPARISON OF
TWO-PHASE FLOW MODELS TO EMPIRICAL DATA
A Senior Honors Thesis
By
PAUL ROBERT MEILLER
Submitted to the Office of Honors Programs 4 Academic Scholarships
Texas A%M University In partial fulfillment of the requirements
For the Designation of
UNIVERSITY UNDERGRADUATE RESEARCH FELLOWS
Approved as to style and content by:
Ron Darby (Fellows Advisor
Edward A. Funkhouser (Executive Director)
April 2000
Group: Engineering
ABSTRACT
Safety Relief Valve Sizing: Comparison of Two-phase
Flow Models to Empirical Data. (April 2000)
Paul Robert Meiller Department of Chemical Engineering
Texas ARM University
Fellows Advisor: Dr. Ron Darby Department of Chemical Engineering
The proper sizing of safety relief valves is an important issue in chemical process safety. Many emergency relief scenarios require consideration of two-phase flow conditions. However, two-phase flow involves complex physics and is the subject of intensive on-going study. The objective of this research is to identify and verify simple yet accurate two-phase flow models which allow the design engineer to predict the mass flux of any given relief scenario. Two contemporary models were considered in this study: The Two-Phase-Homogenous-Equilibrium Model (TPHEM), proposed by the Center for Chemical Process Safety (CCPS), and the Homogenous-Nonequilibrium Model proposed by Fauske. These models were evaluated against steam/water data (both sub-cooled and two-phase entrance) from Sozzi and Sutherland. This research allowed the determination of what conditions were relevant when considering flashing flows, as well as verifying the models accuracy.
ACKNOWLEDGMENTS
The author gratefully and especially acknowledges the assistance of Dr, Ron Darby of thc Chemical Engineering Department, Texas A&M 1Jniversity without whom this research would never have been possible.
Thanks also to Jared Stockton for his work on the shorter Sozzi and Sutherland nozzle 42 data and for his help troubleshooting (and solving) TPHEM problems.
TABLE OF CONTENTS
ABSTRACT.
ACKNOWLEDGMENTS.
ul
1V
TABLE OF CONTENTS . . . . .
LIST OF FIGURES. . V1
LIST OF TABLES .
INTRODUCTION .
BACKGROUND. .
V11
Ideal Nozzle Model, and Single-Phase Flow . . . Physics of Two-Phase Flow.
TWO-PHASE FLOW MODELS.
. . . 3 5
Two-Phase-Homogenous-Equilibrium Model . . . Homogenous Nonequilibrium Model. .
EMPIRICAL DATABASE.
. . . . 9 13
15
METHODOLOGY.
RESULTS AND DISCUSSION . . . . . 22
CONCLUSIONS . . 22
NOMENCLATURE. 31
REFERENCES. 33
VITA 35
LIST OF FIGURES
FIGURE I: SCHEMATIC OF ACTUAL VALVE AND IDEAL NOZZLF . . . . . . . 3
FIGURE 2: SOZZI AND SUTHERLAND NOZZLE CONFIGURATIONS . . . . . 17
FIGURE 3: SOZZI AND SUTHERI. AND NOZZLE ¹2, L = 12. 5 IN . . . . . .
FIGURE 4: SOZZI AND SUTHERLAND NOZZLE ¹2, L = 20. 0 IN . . . . . .
FIGURE 5: SOZZI AND SUTHERLAND NOZZLE ¹2, L = 25. 0 IN . . . . . .
FIGURE 6: SOZZI AND SUTHERLAND NOZZLE ¹2, L = 70 IN . . . . . . . . .
FIGURE 7: SOZZI AND SUTHERLAND NOZZLE ¹3, L = 7. 7 IN . . . . . . . .
FIGURE 8: SOZZI AND SUTHERLAND NOZZLE ¹3, L = 12. 8 IN . . . . . .
FIGURE 9: SOZZI AND SUTHERLAND NOZZLE ¹3, L = 20. 2 IN . . . . . .
FIGURE 10: SOZZI AND SUTHERLAND NOZZLE ¹3, L = 25. 2 IN . . . .
. . . 25
. . . 25
. . . 26
. . . 26
. . . 27
. . . 27
. . . 28
. . . 28
LIST OF TABLES
TABI E 1: TPHEM DENSITY MODELS.
TABLE 2: OVERVIEW OF SOZZI AND SUTHERLAND DATASET. . . . . . . . . . . , . 17
TABLE 3: CASES RUN AND TPHEM PARAMETERS . . . . . . . 1 9
TABLE 4: GOODNESS OF FIT OF SOZZI AND SUTHERLAND NOZZLE ¹2. . 23
TABLE 5: GOODNESS OF FIT OF SOZZI AND SUTHERLAND NOZZLE ¹3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
TABLE 6: OVERALL PERCENT DIFFERENCE FOR MODELS . . . . . . . . 29
INTRODUCTION
IMPORTANCE
An emergency relief system is essential to ensuring the safe operation of process
units. It is almost impossible to guarantee that some sort of overpressuring event will
never occur. Therefore, an emergency relief system stands ready as a "last-resort" safety
device to mediate an overpressure event. To operate properly, the emergency relief
system must be designed properly, and relief valve sizing remains a topic of on-going
study. A properly sized valve is crucial for successful management of emergency
releases. Obviously, an under-sized valve will not aflow adequate mass flow to handle the
pressure build-up, ultimately leading to a tank/reactor rupture. Less evident is the hazard
presented by an over-sized valve. In this case, there may be too high a pressure drop
upstream and downstream of the valve to provide enough pressure difference to keep the
valve open. Thus, an unsteady-state condition results, which can lead to damage to the
valve (from excessive openings and closures). This undesired state of operation is referred
to as valve chatter.
There are two major steps when sizing safety valves. First, the design engineer
must estimate the amount of mass flow the valve must handle to maintain vessel integrity
(e, g. , estimating the amount heat and vapor generated by a runaway reaction). Second, the
engineer must determine the mass flux for a certain valve configuration given the physical
and thermodynamic properties of the system. The required cross-
This thesis follows the style and format of Chemical Engineering Progress.
sectional area (and thus the diameter) of the valve is simply obtained by dividing the mass
flow rate by the mass flux. This paper focuses on a special case of the second step, namely
determining the mass flux through a nozzle under two-phase flow conditions.
Consideration of a two-phase flow system is oflen required in emergency relief
design (1). Two-phase flow, unlike single-phase flow, is poorly understood despite
intensive research efforts. This is not surprising in view of the much greater complexity of
two-phase flow. Successfully modeling two-phase flows has been the objective of a
variety of models, methods, and procedures proposed in the literature (for example, see
Ref. 2, 3, 4, and 5). This paper will attempt to verify two of the most contemporary two-
phase flow models against an extensive two-phase flashing flow database. Systematic
verifications of this type have not been attempted previously. The aim of this research is
to: (1) determine the type and range of system conditions that the models can be
successfully applied to, (2) gain understanding in what types of effects are most important
and which effects can be safely ignored, and (3) ultimately, propose guidelines to aid the
process engineer in successfully and safely applying these models to emergency relief
valve design.
BACKGROUND
SINGLE-PHASE FLOW
By way of introducing the theory underlying mass flux problems, a brief
explanation of single-phase flow is given below (6). Single-phase flow is well
understood, and is modeled by an ideal, adiabatic nozzle (Figure 1 a) modified with a
discharge coefficient, Kn. Note the drastic difference in flow geometry between the actual
valve (Figure lb) and the ideal nozzle. The discharge coefficient can range from 0. 6-0. 7
for liquids to close to unity for gases. The various valve manulacturers publish empirical
discharge coeflicients for single-phase flows only in the so-called "Red Book" (7). No
data on two-phase flow discharge coeflicients are routinely published.
P„
EDUCTOR
GUIDE
DISC HOLDER
DISC
ADJUSTING RING Pb
ADJ. RING
PRIMARY ORFICE
NOZZLE
II ~
INLET NECK
THREADS
a) ideal Nozzle b) Typical Spring-Loaded Relief Valve
Figure I: Schematic of Actual Valve and Ideal Nozzle
To calculate the ideal mass flux, G„, ~a, the following information is required: stagnation
(upstream) pressure, P&„' exit pressure, P„; and the relationship between the fluid density
and pressure. The actual mass flux, G, @„„ is obtained by multiplying by Ko.
P
~-' = K' P'-** = K" P ~2
I-— p. P
An important complication arises in gas/vapor flows. Due to the compressible nature of
gases/vapors these systems can reach a maximum flux or "choke" point where the local
velocity equals the speed of sound (i. e. the sonic velocity). The pressure that corresponds
to the maximum velocity is referred to as the choke pressure or P, . Once the choke
pressure is reached, any further pressure drop that occurs in the system does not affect the
mass flux calculation. This is an important consideration as it explains how the ideal
nozzle results are in such good agreement with empirical gas flows. Most releases involve
large pressure drops that tend to choke very quickly. If the choke pressure occurs at the
nozzle exit, then the rest of the valve becomes irrelevant. Note that the valve nozzle itself
closely resembles the ideal nozzle. In practice, when using Eq. 1 to calculate the
maximum mass flux, the exit pressure is lowered until a maximum in G„, a is obtained.
This final pressure is the choke pressure of the system.
TWO-PHASE FLOW
Although two-phase flow is substantially more complex, the basic approach to
sizing relief valves is the same as single-phase flows: choosing the appropriate model for
the pressure-density relationship so Eq. I can be evaluated, and determining the proper
discharge coefficient to use. The discharge coeflicient is of considerable concern since
there is presently no published Kii data for two-phase flow through relief valves. In order
to select an appropriate model, assumptions must be made with regard to a number of
factors which are discussed by Darby (1). The more important considerations are
summarized below.
Phase Distribution
Differing relative flow rates and relative amounts of the two phases gives rise to a
variety of flow regimes. Although a wide range of regimes are possible, the typical
assumption for relief valve scenarios is homogeneous flow, where the gas and liquid
phases are uniformly mixed. A common-day example of this is the flow out of a well-
shook champagne bottle. When analyzing relief valves, the common assumption is that
the two-phase mixture can be represented as a "pseudo- homogeneous single-phase fluid".
The physical properties of this "homogenous fluid" are some appropriate weighted
average of the properties of the separate phases. Both models discussed in this paper
make use of the pseudo-homogeneous fluid assumption.
Thermodynamic State
How the thermodynamic state of the fluid changes across the valve body is a major
consideration that must be accounted for. A number of different cases can exist. A
"frozen" system is one where the quality (vapor mass fraction) of the fluid is not a
function of pressure and therefore does not change (i. e. the relative amounts of gas/vapor
and liquid remain constant). A cold air/water mixture is an example. "Flashing" systems
(e. g. water/steam) do change quality as pressure drops across the valve and the liquid
evaporates or boils. "Saturated" and "slightly sub-cooled" systems are a sub-set of the
"flashing" condition. In the "saturated" case, the initial saturated liquid undergoes
additional flashing through the valve. "Slightly sub-cooled" systems enter the nozzle as an
all-liquid flow and do not flash until afler the saturation pressure has been reached. It is of
extreme importance for any model to account for these different conditions.
Thermodynamic Equilibrium
While it is tempting to simply assume local thermodynamic equilibrium when
analyzing flashing flows, this assumption ignores the rate processes involved in the
flashing process (i. e. nucleation site formation, heat transfer involved in bubble growth,
etc. ). These rate processes restdt in an effective delay in the initial generation of vapor
after the saturation pressure has been reached. In systems where flashing has already
generated a significant amount of vapor, the assumption of local thermodynamic
equilibrium appears to be valid. Otherwise, when dealing with "saturated", "slightly sub-
cooled", or "slightly superheated" systems (i. e. systems with low qualities) the non-
equilibrium nature of flashing must be considered.
Mechanical /qonequilibriutn (Slip)
As the pressure drops across the valve body, the gas/vapor phase will expand (i. e.
density decreases, specific volume increases) while the liquid phase density remains
constant. Thus, the gas phase will accelerate relative to the liquid phase, and a so-called
"slip" condition is created. Slip affects the local volume fraction of gas/vapor or the
gas/vapor "hold-up". Slip effectively increases the liquid "hold-up" for a given location
within the system. General situations where slip might be important include: low quality
flows (qualities less than 1'/0), &ozen systems, and systems with large pressure gradients.
Choked Conditions
Like single-phase flow, two-phase flows reach a maximum mass flux when the
sonic velocity is reached. Unlike single-phase flow, however, the speed of sound is less
clearly defined. Still, the same basic technique can be used, namely lowering P„ to
determine the maximum flux. Flashing flows will choke at much higher exit pressures
than single-phase gas systems. For comparison, most single phase gas/vapor systems will
choke at roughly half of the inlet pressure while two-phase systems may choke at up to
90'/o of the inlet pressure. Frozen flows may choke over a much wider range of pressures,
but generally at lower pressures than flashing flows.
Fluid Composition
In addition to all the above considerations, the composition of the fluid can be very
important. Many relief systems are installed on reactors that contain a complex mixture of
chemicals. These chemicals could have either low or high viscosity. They could behave as
a Newtonian, non-Newtonian, or even viscoelastic fluid. "Foamy" mixtures are also a
concern (8). Foaminess is especially hard to predict because of its extreme sensitivity to
surface tension. Each of these factors must be considered to properly determine the fluid
pressure-density relationship. The affect many of these factors have on two-phase flows is
poorly understood and the subject of on-going research.
TWO-PHASE FLOW MODELS
Many computational methods and models for predicting two-phase mass flux
through an ideal nozzle are available in the literature (I). Many of these are designed to
apply only to a limited number of the special cases discussed above. These models may be
split into two rough categories: "two-fluid" and "homogenous". The "two-fluid" models
rely on microscopic conservation equations of mass, energy, and momentum. The
balances are generally applied locally to each phase and a model for inter-phase transport
is also included. As might be expected, these models are quite computer intensive, require
a large amount of system data, and can be applied to only limited cases. These models are
more appropriate for detailed mechanism investigations then engineering design
calculations. "Homogenous" models, on the other hand, make the "pseudo-homogenous-
single-phase-fluid" assumption as discussed previously. This greatly simplifies the
calculations and drastically reduces the amount of system information required. As such,
these models are far easier to apply to relief system design. In addition, it is not at all
evident that "two-fluid" models provide more accurate results than the "homogenous"
models. This paper evaluates and verifies two recent "homogenous" models that are able
to handle a wide range of the cases discussed above. A detailed discussion of each model
follows.
Two-Phase-Homogenous-Equi librium Model
The Two-Phase-Homogenous-Equilibrium-Model (TPHEM) is a computer model
described recently by the Center for Chemical Process Safety (CCPS) (5). TPHEM has
10
two different calculation methods that are of interest: the nozzle case, and the pipe case. In
the nozzle case, TPHEM uses numerical integration to solve the ideal nozzle equation:
'r dP1 Gnozza = pn 2 f— J pj
(2)
where G„„. ~, is the mass flux through the nozzle, P, is the (upstream) stagnation pressure,
P„ is the pressure at the nozzle exit, and p is the local two-phase fluid density (p„ is the
exit density). The integration schemes are based on the papers of Simpson (9, 10). The
local two-phase density is calculated by making the "pseudo-homogenous-single-phase-
fluid" assumption. The pseudo-fluid density is related to the densities of the gas and liquid
phases by
P = + Po+(I-~)Pr.
where ct is the volume fraction of the gas phase. The TPHEM model incorporates a
number of possible models for the pressure-density relationship (Table I ). Depending on
the model chosen, the user must input information from one, two, or three points in the
system (referred to as point A, B, and C). For each point, the pressure, the gas/vapor
quality (i. e. mass fraction), the gas/vapor density, and the liquid density must be given.
The program evaluates the model parameters from the data entered and then numerically
integrates the mass flux integral from stagnation to discharge pressure. If the flow is
choked, as is often the case, a maximum flux will be found before the discharge pressure
is reached at the critical choke pressure. The one-parameter model ("A" in Table I) is
equivalent to the Omega model proposed by Leung (1 I).
MODEL POINTS EQUATION
A (p, /p — 1) = a(P, / P — 1)
B (p, /p — 1) = a(P, /P — 1)'
C (p, /p — 1) = a[(P, /P)' — 1]
(p. / p — 1) = a(P, / P — 1) + b(P. / p — 1)'
x =a, +a, P
po =b, P '
pL =c +cP
x = a, +a, P+a, P
(P, „ /Po — 1) = b. [(P, /P)" — 1]
p L c p + P + c, P
Table 1: TPHEM Density Models
TPHEM, Nozzle case also has the capability to account for both mechanical (slip)
and thermal nonequilibrium. There are many slip models and little or no experimental
confirmation of which is most suitable. Nevertheless, TPHEM includes two methods
which require the user to either input the slip ratio (S = Vo / VL) directly or to input a k,
parameter by which S is calculated;
S= 1-x+x (4)
12
Nonequilibrium is included by replacing the local equilibrium quality (x) by a
"nonequilibrium" quality (x~s) where x~s & x:
xse = x, +k„s(x — x, )
and ktts is an empirical parameter (input by the user), and x, is the quality at the
stagnation condition (in this equation, x, is zero for subcooled inlet conditions). Simpson
asserts that a k~= 1 gives results comparable to the Homogenous Nonequilibrium Model
(see below).
As stated previously, TPHEM also has the capability of analyzing pipe flows. To
calculate the mass flux, TPHEM, Pipe replaces Eq. 2 with Eq. 6 which is a rearrangement
of the Bernoulli's equation for pipe flow:
I — pdP+gAZp', „,
P
ln(p, /p. )+(K, +1)/2
1/2
(6)
where g is the acceleration of gravity, AZ is the elevation change, and Kris the friction
loss (/0). TPHEM, Pipe includes the effect of friction loss in the pipe and fittings, but
does not include terms for slip or nonequilibrium.
13
Homogenous Xonequilibrium Model
This model has evolved over time and has been recently described by Fauske (3):
)/2
Gc
G, 1+K,
N~ = — ' + for L&L -(GJ . , (g)
for L&L,
P P
= J2P(P. — P, ) (10)
h GLo G, = vo) ~T. CP, L.
G, =J2p (P, — P) (12)
where,
Gi
NNa =
Ki-
two-phase mass flux
GssM = equilibrium rate (see below)
nonequilibrium parameter
loss coefficient = 4fL/D (for straight pipe)
14
L, = equilibrium length = 10 cm
pc, = liquid density at stagnation condition
P, = stagnation pressure
P, = saturation pressure
hoi, = heat of vaporization at stagnation conditions
voip = specific volume of gas minus specific volume of liquid at stagnation
conditions
T, = stagnation temperature
Cs t = specific heat of liquid at stagnation conditions
Pt = discharge pressure (choke pressure if flow is choked)
G, represents the liquid flow from stagnation to saturation for subcooled inlets. Gi
represents the critical (choked) mass flux resulting from the phase change, and Gs is the
liquid flow from saturation to discharge pressure. For choked flows, Pt is replaced by the
choke pressure, P, . The nonequilibrium parameter, NNs, represents the delayed flashing
for lengths less than 10 cm. The 10 cm criteria comes from studies by Fauske that show
that flashing is normally completed within 10 cm of the inlet (12). Fauske's model does
not include the effect of slip.
15
EMPIRICAL DATABASE
The great majority of two-phase flow data comes from studies conducted by the
nuclear power industry. Most of this research was driven by work on boiling water
reactors and predictions of flow rates in boiler tubes and broken pipes or leaks. One EPRI
Report (13) compiled the available data sources and then screened the sources for
inconsistent and incomplete data sets. Thus, this report gives a comprehensive and reliable
database for steam/water two-phase flow in pipes and nozzles. The Sozzi and Sutherland
database tabulated in this report provides the most consistent and useful data (/4). This
database includes a number of different nozzle configurations each attached to a number
of different lengths of exit piping. This paper considered nozzles ¹2 and ¹3. Nozzle ¹2 has
a well-rounded entrance while nozzle ¹3 has a square entrance. Figure 2 shows the
dimensions of the two configurations and Table 2 lists the length, number of data points,
and run conditions (range of qualities and range of stagnation pressures) for each dataset
considered in this paper. These datasets represent the "pipe" cases (i. e. those cases with
long runs of exit piping). Darby, et. al. has used a methodology very similar to this paper's
to compare both TPHEM and HNE to six shorter "nozzle" cases (6). The "nozzle" cases
were nozzle ¹2 with the following exit length: 0 in, 1. 5 in, 2. 5 in, 4. 5 in, 7. 5 in, and 9 in.
For each run, the following data were reported: stagnation pressure, P„stagnation
enthalpy, H, ; stagnation quality, x, ; mass flux, G„' critical pressure, P, ; and critical
quality, x, . The Sozzi and Sutherland data were selected because of the large number of
datasets, usefulness of the range of run conditions, and the consistency of the data. The
16
range covers slightly sub-cooled to low quality entrance conditions, thus bracketing the
saturation condition. Systems close to saturation are the most difficult to model.
1. 75 in.
L
0. 5 in. 0. 5 in
a) Nozzle ¹2 b) Nozzle ¹3 Figure 2: Sozzi and Sutherland Nozzle Configurations
LENGTH ¹ OF Xo RANGE P0 RANGE (in. ) POINTS ( - ) (psig)
NOZZLE ¹2 — Rounded Entrance
12. 5
20. 0
19
13
-0. 0023 to 0. 0043
-0. 0020 to 0. 0020
817 to 985
831 to 975
25. 0 96 -0. 0043 to 0. 0043 845 to 1023
70. 0 81 -0, 0043 to 0. 0034 877 to 988
NOZZLE ¹3— Square Entran ce
7. 7 24 -0. 0020 to 0. 0050 863 to 996
12. 8 24 -0. 0020 to 0. 0040 874 to 999
20. 2 17 -0. 0030 to 0. 0040 879 to 1013
25. 2 23 -0. 0020 to 0. 0040 867 to 1009
Table 2: Overview of Sozzi and Sutherland Database
METHODOLOGY
The eight datasets listed in Table 2 were used as inputs for both the TPHEM and
HNE models. The empirical mass flux values were compared to the calculated mass flux
for various cases of each of these models. In an effort to determine which parameters and
conditions were important and which could be ignored, the datasets were run against a
total of seven different cases. Table 3 lists these cases and also lists the numerical value of
the switches used in the TPHEM cases. Refer to the CCPS guidelines for detailed
instructions on using TPHEM (5).
A number of assumptions were needed to apply the models to the Sozzi and
Sutherland database. The quality (vapor mass
fi
actio) was calculated by assuming the
flow through the nozzle or pipe is isenthalpic (which gives results very close to the
isentropic assumption). Quality is thus defined as
H„— H, x = Ho -H„ (l3)
wllel'e Hp is the initial enthalpy as reported in the dataset, and Ho and HL refer to the
enthalpy of the gas phase and liquid phase, respectively, at the local pressure. TPHEM
requires the entrance and exit pressures. Eq. 13, along with given thermodynamic data and
steam table properties, was used to generate the three additional data points required by
TPHEM. The first data point, point A, was taken to be at saturation conditions. The
pressures at points B and C were taken to be 0. 75 Px and 0. 50 PA, respectively. It was
CASK DESCRIPTION TPHEM PARAMETER'
IU IC IPTS IV
TPHEM, Nozzle w/ friction
TPHEM, Nozzle w/o friction
TPHEM, Nozzle w/ slip
TPHEM, Nozzle, KNE Fit
1. 5
11 Variable
TPHEM, Pipe w/ friction -3
HNE, w/ friction
HNE, w/o friction
TPHEM Parameters: IU IC
Units Case. Option (1) give flow rate output. (3) also gives flow rate output and in addition activates INES, the advanced option flag.
IPTS — Model and number of data states. (3) corresponds to model F in Table I and three data points.
IV — Input Options. (I) is simple non-viscous input and (-3) is pipe input without viscosity correction
INES — Advanced Options. Option (2) allows the user to input a slip ratio, S. (11) is used to input a KNE value for nonequilibrium corrections.
X — Advanced Options Value. The value for either S or kiis depending on the value of INES (see above).
These cases were so grossly inaccurate they were eliminated from the study. The KNE fit case attempted to find the value of kNs that would fit the empirical result. Thus, the kNs value varied from 0 to 75.
Table 3: Cases Run and TPHEM Parameters
20
then straightforward to determine the other required data, local qualities, gas phase
densities, and liquid phase densities, by using Eq. 13 and the steam tables. The steam table
used was a MicrosoA Excel plug-in based on the 1967 ASME code. A consistent method
for calculating the friction loss was also needed. Several methods were tried and rejected
before a final choice was made. The TPHEM Pipe case calls for a roughness factor. The
pipe was assumed to have a roughness of 0. 0004 mm, typical of stainless steel. To
maintain consistency with the other cases, this roughness value was used to calculate an
equivalent loss coefficient, Kr which is required for the TPHEM, nozzle case and HNE.
The Churchhill equation was used to relate the friction factor to the roughness and
Reynolds number (15).
N, . (A+ B)"' (14)
2. 457 ln 1
7 0. 278 (15)
37, 530 (16)
DVp DG. V lti.
(17)
and
4fL D
(18)
21
where D is the diameter of the straight pipe, G, is the mass flux as taken from the dataset,
lit. is the viscosity of water (taken to be constant at 1 cP), and L is the length of the
straight pipe. This method was simple to apply and gave consistent results. For nozzle ¹3, an entrance loss of 0. 4 was added to the calculated value from Eq. 18. Finally, to correctly
apply the HNE model a value for the choke pressure, P, was needed for Eq. 12. This value
was taken from the Sozzi database when provided, otherwise the value was taken &om the
output of the TPHEM, Nozzle w/ &iction case.
The TPHEM, KNE fit case was different from the other cases in that it required an
iterative procedure to arrive at an answer. The aim of the KNE fit test was to find the kiis
value that would most closely match the empirical results. ktis values from 0 to 75 were
input. This test was later eliminated for the "pipe" cases in this study for reasons discussed
below.
With the exception of the KNE fit test, all other cases result in a calculated mass
flux, G„i, . TPHEM also gave final values for critical pressure and quality. The calculated
mass flux value was compared with the empirical value, G, b„ for each data point. For
each dataset a value for the average G„i JG, s, was calculated as well as the standard
deviation. These results are shown in Table 4. A plot of the dimensionless mass flux (G, )
vs. stagnation quality (x„) was also produced, where
Gc
~p„p. (1 g)
Refer to Figures 3 - 10 for these plots.
22
RESULTS AND DISCUSSION
Several of the cases listed in Table 3 were found to be inappropriate and were
eliminated &om the study. The TPHEM, nozzle without friction and the HNE without
friction did not account for the frictional loss and therefore over-predicted the mass flux
quite significantly (greater than 205'0). This was especially hue for the longer pipes.
TPHEM, nozzle with slip was shown to return results almost identical to the TPHEM,
Nozzle without friction case. This led to the conclusion that, in flashing systems, slip
effects are negligible compared to the flashing effects. The TPHEM, KNE fit test was
shown to be useless for longer nozzles. In sub-cooled and saturated cases (and even a few
two-phase cases) the TPHEM model would become insensitive to the k~ value and would
exhibit a limiting behavior at mass fluxes much higher than the empirical data (greater
than 15'10) for very large values of kzs. It was shown that in cases exhibiting this limiting
behavior any as value over 75 would have negligible effect on the output. The results
from all of the above cases were deemed physically meaningless and are not reported.
The results of the three remaining cases for nozzle ¹2 and nozzle ¹3 are shown in
Table 4 and Table 5, respectively. The mass flux vs. quality plots are contained in Figures
3-10.
23
DATASET STANDARD ' ' '"a DEVIATION
NOZZLE ¹2 — Rounded Entrance — L = 12. 5 in.
TPHEM, Nozzle w/ &iction
TPHEM, Pipe w/ &icnon
HNE, w/ friction
0. 904
0. 935
1. 058
0. 039
0. 045
0. 059
NOZZLE ¹2 — Rounded Entrance — L 20. 0 in.
TPHEM, Nozzle w/ &iction
TPHEM, Pipe w/ &iction
HNE, w/ friction
0. 884
0. 931
1. 038
0. 066
0. 071
0. 041
NOZZLE ¹2 — Rounded Entrance — L = 25. 0 in.
TPHEM, Nozzle w/ friction
TPHEM, Pipe w/ friction
HNE, w/ friction
0. 938
0. 997
1. ] 09
0. 061
0. 083
0. 077
NOZZLE ¹2 — Rounded Entrance — L = 70. 0 in.
TPHEM, Nozzle w/ &lotion
TPHEM, Pipe w/ friction
HNE, w/ friction
0. 973
1. 008
1. 006
0. 063
0. 074
0. 076
Note: TPHEM, Pipe used a roughness factor of s=0, 0004 mm and other cases used an equivalent
Kr (see /v/erhodologv).
Table 4: Goodness of Fit of Sozzi and Sutherland Nozzle ¹2
24
DATASET STANDARD DEVIATION
NOZZLE ¹3 — Square Entrance — L = 7. 7 in.
TPHEM, Nozzle w/ friction 0. 853 0. 051
TPHEM, Pipe w/ friction
HNE, w/ friction
0. 929
1. 004
0. 057
0. 041
NOZZLE ¹3 — Square Entre nce — L = 12. 8 in.
TPHEM, Nozzle w/ friction
TPHEM, Pipe w/ friction
HNE, w/ friction
0. 941
1. 054
1. 108
0. 143
0. 075
0. 095
NOZZLE ¹3 — Square Entrance — L = 20. 2 in.
TPHEM, Nozzle w/ friction
TPHEM, Pipe w/ friction
HNE, w/ friction
0. 951
1. 030
1. 037
0. 141
0. 072
0. 079
NOZZLE ¹3 — Square Entra nce — L = 25. 2 in.
TPHEM, Nozzle w/ friction 0. 968 0. 165
TPHEM, Pipe w/ friction
HNE, w/ friction
1. 1061
1. 072
0. 140
0. 099
Note: TPHEM, Pipe used a roughness factor of a=0. 0004 mm and other cases used an
equivalent Kr plus 0. 4 for entrance loss (see Methodology).
Table 5: Goodness of Fit of Sozzi and Sutherland Nozzle ¹3
25
0. 8
0. 7
D Data
~ HNE, Kf=4fUD
0. 6
~M 0. 5
~ R e TPHEM Pipe
e=0. 0004
~ TPHEM, Kf=4tUD
0. 4
0. 3
Q g
j
0. 2 -0. 003 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003 0. 004 0. 005
Xo
Figure 3: Sozzi and Sutherland Nozzle ¹2, L = 12. 5 in
0. 8
0. 7 tt Data
a HNE, Kf=4fUD
0. 6 e TPHEM Pipe e=0. 0004
~M 0. 5 ~ TPHEM, Kf=4fUD
0. 4
0. 3
0. 2 -0. 003 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003
Xo
Figure 4: Sozzi and Sutherland Nozzle ¹2, L = 20. 0 in
0. 8
0. 7 D Dafa
a HNE, Kf=4fL/D
0. 6 D Da+
4 ac 0
e TPHEM Pipe e=0. 0004
~ TPHEM, Kf=4fL/D
0. 4
0. 3
'ID gi ~ ~ E ~
0. 2 -0. 005 -0. 004 -0. 003 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003 0. 004 0. 005
Xo
Figure 5: Sozzi and Sutherland Nozzle ¹2, L = 20. 0 in
0. 8
0. 7 ~ HNE, Kf=afUD
0. 6
~M0. 5
0. 4
'a D D
~ cpa D
e TPHEM Rpe e=0 0004
~ TPHEM, Kf=4
0. 3
0. 2 -0. 005 -0. 004 -0. 003 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003 0. 004 0. 005
Xo
Figure 6: Sozzi and Sutherland Nozzle ¹2, L = 70. 0 in
27
0. 8
0. 7 e HNE, Kf=4tUD
0. 6
~o 0. 5
D
ff
~ TPHEM Pipe eHt 0004
~ TPHEM, Kf=4fU
0. 4
0. 3
~ , 'Q D
~ ~ ~ 4
~ ', ~ ~ ee
0. 2 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003 0. 004 0. 005 0. 006
Xo
Figure 7: Sozzi and Sutherland Nozzle ¹3, L = 7. 7 in
0. 8
0. 7 l ~ HNE Kt=etUD
0. 6
~o 0. 5 ~ TPHEM, Kf&fUD
0. 4
0. 3 I y, 'i
0. 2 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003 0. 004
Xo
Figure 8: Sozzi and Sutherland Nozzle ¹3, L = 12. 8 in
28
0. 8
0. 7 ~ HNE, Kf=4fUD
0. 6
~o 0. 5
4 TPHEM Pipe ~. 0004
~ TPHEM, Kf=4
0. 4 D
0. 3
0. 2 -0. 004 -0. 003 -0. 002 -0. 001 0. 000 0. 001 0. 002 0. 003 0. 004
Xo
Figure 9: Sozzi and Sutherland Nozzle ¹3, L = 20. 2 in
0. 8—
0. 7 4 HNE, Kf4fUD
0. 6
g 0. 5
4 TPHEM Pee ~. 0004
~ TPHEM, Kf=4IUD
0. 4
0. 3
0. 2 -0. 002 0. 000 0. 002 0. 004
Xo 0. 006
Figure 10: Sozzi and Sutherland Nozzle ¹3, L = 25. 2 in
29
Inspection of the tables and the plots reveals some general trends. First, all cases
seem to improve in accuracy as the exit piping length increases. This might be due to
some non-equilibrium effects not accounted for in the models. Second, in all cases two-
phase entrance conditions gave better agreement than sub-cooled conditions. All cases
tend to over-predict the mass flux for sub-cooled entrance. This inaccuracy worsens as the
degree of subcooling increases. Therefore, it would seem both TPHEM and HNE are not
accounting for all the vapor generation under sub-cooled conditions. As can be seen in
Table 6, overall, TPHEM, Pipe was the most accurate model based on average fit, but it
gave the highest deviation (i. e. scatter). HNE tended to over-predict the mass flux under
all conditions, while the TPHEM, nozzle case tended to under predict.
MODEL OVERALL PERCENT
DIFFERENCE
STANDARD DEVIATION
TPHEM, nozzle w/ friction -7. 3% (under) 4. 2%
TPHEM, pipe w/ friction 0. 1% (over) 6. 5%
HNE, w/ &iction -5. 4% (over) 4. 1%
Table 6: Overall Percent Difference for Models
30
CONCLUSIONS
As previously stated, an understanding of what conditions are important and which
can be neglected is vital to applying these models correctly. The results of this paper have
shed some light on this area. It was shown that slip is completely dominated by flashing
effects in both two-phase and sub-cooled conditions for long nozzles and pipes. Thus, it
can be safely eliminated for flashing flows only (for Irozen flows slip can be very
important and should not be eliminated). As previously stated by Fauske, nonequilibrium
effects are not appreciable when the exit piping is much greater than 10 cm (1 1). Friction
loss is a very important consideration (especially as the exit piping length increases) and
should not be neglected.
The above results show that all three models give results in good agreement with
empirical data over the entire range of inlet quality and piping length. As such, these
models are acceptable for most design calculations especially relief valve sizing.
However, in order to obtain these results a complete knowledge of the fluid's
thermodynamic and phase state must be known. In a design problem, the ideal nozzle
model would need to corrected by adding a discharge coefficient. Although there is only
sparse amounts of two-phase flow through valve data available, what little there is
suggests that discharge coefficients close to unity are acceptable (6). Regardless, to better
predict this two-phase discharge coefficient further research is needed to investigate the
relationship between the ideal nozzle and actual valve geometry.
31
NOMENCLATURE
a, b, c
Cp, t
G
hollo
Kr
ks
L
L,
N„,
XNE
empirical parameters for TPHEM density models in Table 1
specific heat of liquid at stagnation conditions (ft lbi/ Ibm 'F or N m / kg "C)
straight pipe diameter (ft or m)
friction factor ( — )
acceleration of gravity = 32. 17 fl / s or 9. 8 m / s
mass flux (Ib~ / fl s or kg / m s)
dimensionless mass flux = Gp/ (Po po) ( )
heat of vaporization at stagnation conditions (ft lbr / lbrp or N m / kg)
discharge coefficient ( — )
friction loss coefficient ( - )
TPHEM nonequilibrium parameter ( - )
TPHEM slip parameter ( - )
nozzle length (ft or m)
equilibrium length for HNE model = 10 cm
nonequilibrium parameter defined by Eq. 7 and 8 ( — )
pressure (lbr/ ft or Pa)
slip ratio, ration of gas phase velocity to liquid phase velocity ( - )
temperature ('F or 'C)
quality, mass fraction of gas or vapor in mixture ( - )
TPHEM nonequilibrium quality
velocity(ft/ s or m/ s)
32
hZ elevation change (ft or m)
Cxreek
volume fraction of gas phase in mixture ( - )
roughness factor ( - ) specific
volum(ft /lb or m /kg)
ooio specific volume of gas minus specific volume of liquid at stagnation conditions (fr'/lb or m /kg)
density(lb /ft orkg/m)
viscosity(lb /As or kg/ms)
Subscripts
critical (choked) state
gas or vapor phase
liquid phase
discharge (exit) state.
stagnation (upstream) state
saturated state
33
REFERENCES
Darby, R. , "Pressure Relief Device Sizing: What's Known and What's Not", presented at the 1998 Plant Process Safety Symposium, Houston, TX (October 26- 27, 1998).
Fisher, H. G. , H. S. Forrest, S. S. Grossel, J. E. Huff, A. R. Muller, J. A. Noronha, D. A. Shaw, and B. J. Tilley, "Emergency Relief System Design Using DIERS Technology", AIChE, NY (1992).
Fauske, H. K. , "Determine Two-Phase Flows During Releases", Chemical Engineering Progress, pp. 55-58 (February 1999).
Darby, R. , "Viscous Two-Phase Flow in Relief Valves", Phase I Report to DIERS/AIChE (November 1997).
Center for Chemical Process Safety, "Guidelines for Pressure Relief and Effluent Handling Systems", AIChE, NY (1998).
Darby, R. , P. Meiller, J. Stockton, "Relief Sizing for Two-Phase Flow", presented to the 34 Annual Loss Prevention Symposium, Atlanta, GA (March 5-9, 2000).
National Board of Boiler and Pressure Vessel Inspectors, "Pressure Relief Device Certifications", Columbus, OH (1997).
Fauske, H. K. , "Properly Size Vents for Nonreactive and Reactive Chemicals", Chemical Engineering Progress, pp. 17-29 (February 2000).
Simpson, L. L. , "Estimate Two-Phase Flow in Safety Devices", Chemical Engineering, pp. 98-102 (August 1991).
10. Simpson, L. L. , "Navigating the Two-Phase Maze". Proceedings of the International Symposium on Runaway Reactions and Pressure Relief Design G. A Melhem and H. G. Fisher, eds. AIChE/DIERS, NY (1995).
11. Leung, J. C. , "Easily Size Relief Devices and Piping for Two-Phase Flow", Chemical Engineering Progress, pp. 28-50. (December 1996),
12. Fauske, H. K. , "Flashing Flows or: Some Guidelines for Emergency Releases", Plant Operations Progress, v. 4, no. 3, pp. 132-134 (1985).
34
13. Bannerjee, S. , and S. Behring, "A Qualified Database for the Critical Flow of Water", EPRI Report NP-4556 (May 1986).
14. Sozzi, G. L. and W. A. Sutherland, "Critical Flow of Saturated and Subcooled Water at High Pressure", Report NEDO-13418, General Electric Company, San Jose, CA (July 1975).
15. Darby, R. Chemical Engineering Fluid Mechanics. Marcel Dekker Press. New York (1996).
35
VITA
Paul Robert Meiller is a senior undergraduate chemical engineering major at Texas
A&M University at College Station, Texas. As an Honors Undergraduate Research
Fellow, he studied two-phase fluid flow under Dr. Ron Darby, Chemical Engineering
Department, Texas A&M. Meiller shares one publication with Dr. Darby and Jared
Stockton. He has two summers of industry experience as a Process Engineering Intern
with Solutia, Inc, and, in the summer of 2000, he will travel to Ludwigshafen, Germany to
participate in BASF's Summer Internship Program. At Texas A&M, Meiller has been
Vice-president for the Student Chapter of the American Institute of Chemical Engineers
(AIChE) as well as holding numerous other leadership positions on campus. He will
graduate with honors from A&M in the fall of 2000, and he plans to take a position in the
chemical industry for several years before pursuing any advanced degrees.
Meiller's permanently resides at 1426 North Road, Lake Jackson, TX 77566.